Strong Randomness Properties of (Hyper-)Graphs Generated by Simple Hash Functions Martin Aum¨ uller Technische Universit¨ at Ilmenau, Germany AofA’15 Strobl, June 8, 2015 Joint work with Martin Dietzfelbinger and Philipp Woelfel. M. Aum¨ uller Graphs Generated by Simple Hash Functions 1/17
Example: Cuckoo Hashing (Pagh/Rodler, 2001/2004) A hashing-based implementation of the dictionary data type. Setting: set S ⊆ U of n keys two tables T 1 [0 .. m − 1] and T 2 [0 .. m − 1], m ≥ (1 + ε ) n two (hash) functions h 1 , h 2 with h i : U → [ m ] Rules: each table cell can hold exactly one key a key x must be stored either in T 1 [ h 1 ( x )] or T 2 [ h 2 ( x )] (fast lookups and deletions!) Definition If S can be stored according to these rules, we call ( h 1 , h 2 ) suitable for S . M. Aum¨ uller Graphs Generated by Simple Hash Functions 2/17
Example: Cuckoo Hashing (Pagh/Rodler, 2001/2004) A hashing-based implementation of the dictionary data type. Setting: set S ⊆ U of n keys two tables T 1 [0 .. m − 1] and T 2 [0 .. m − 1], m ≥ (1 + ε ) n two (hash) functions h 1 , h 2 with h i : U → [ m ] Rules: each table cell can hold exactly one key a key x must be stored either in T 1 [ h 1 ( x )] or T 2 [ h 2 ( x )] (fast lookups and deletions!) Definition If S can be stored according to these rules, we call ( h 1 , h 2 ) suitable for S . M. Aum¨ uller Graphs Generated by Simple Hash Functions 2/17
Improving Cuckoo Hashing: Stash Original Analysis: ( h 1 , h 2 ) unsuitable with probability O (1 / n ). In fact: Θ(1 / n ) (Schellbach ’09, Drmota/Kutzelnigg ’12) (Kirsch/Mitzenmacher/Wieder ’08): Θ(1 / n ) is too large. Proposal: Can put up to s = O (1) keys into additional storage Theorem (K/M/W ’08) Let S ⊆ U with | S | = n . If ( h 1 , h 2 ) are fully random , then Pr(( h 1 , h 2 ) unsuitable for S with stash size s ) = O (1 / n s +1 ) . Again: Θ(1 / n s +1 ). (Kutzelnigg ’10) M. Aum¨ uller Graphs Generated by Simple Hash Functions 3/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) 3 more keys than table cells ⇒ at least 3 keys must be put into stash M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) 3 more keys than table cells ⇒ at least 3 keys must be put into stash M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) 3 more keys than table cells ⇒ at least 3 keys must be put into stash M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) 3 more keys than table cells ⇒ at least 3 keys must be put into stash M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Excess (Janson et al. ’93): #edges - #vertices (Here: 3) 3 more keys than table cells ⇒ at least 3 keys must be put into stash Minimal “bad subgraph”: a MOS s . (Example: s = 2.) M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
Analysis of Cuckoo Hashing with a Stash What is a criteria for ( h 1 , h 2 ) being unsuitable for stash size s ? Tool: Cuckoo graph G ( S , h 1 , h 2 ) (Devroye/Morin ’03) Theorem (K/M/W ’08) Let ( V ′ , E ′ ) consists of all connected components of G ( S , h 1 , h 2 ) having more than one cycle. Then Stash size = | E ′ | − | V ′ | . M. Aum¨ uller Graphs Generated by Simple Hash Functions 4/17
The Quest M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . Replace fully random hash functions by an explicit, efficient construction of hash functions. M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . Replace fully random hash functions by an explicit, efficient construction of hash functions. “Simple hash functions that work in as many applications as possible” M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . Replace fully random hash functions by an explicit, efficient construction of hash functions. “Simple hash functions that work in as many applications as possible” Other recent approaches, e. g., Thorup/Pˇ atra¸ scu ’11, Reingold/Rothblum/Wieder ’14 M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . Replace fully random hash functions by an explicit, efficient construction of hash functions. “Simple hash functions that work in as many applications as possible” Other recent approaches, e. g., Thorup/Pˇ atra¸ scu ’11, Reingold/Rothblum/Wieder ’14 Focus on hashing-based algorithms and data structures that allow good enough bounds via first-moment method (C.H. [stash], generalized C.H., load balancing, ...) M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
The Quest Analysis well understood when hash functions are fully random . Replace fully random hash functions by an explicit, efficient construction of hash functions. “Simple hash functions that work in as many applications as possible” Other recent approaches, e. g., Thorup/Pˇ atra¸ scu ’11, Reingold/Rothblum/Wieder ’14 Focus on hashing-based algorithms and data structures that allow good enough bounds via first-moment method (C.H. [stash], generalized C.H., load balancing, ...) Generic approach? M. Aum¨ uller Graphs Generated by Simple Hash Functions 5/17
Key Ingredient: Linear Functions h ( x ) = (( a · x + b ) mod p ) mod m , where p ≥ | U | is a prime, and a and b are chosen uniformly at random from { 0 , . . . , p − 1 } . → very simple structure! (Remark: This function is 2-wise independent, i. e., for any pair x , y ∈ U , x � = y , h ( x ) and h ( y ) are fully random.) M. Aum¨ uller Graphs Generated by Simple Hash Functions 6/17
The Hash Class (Version for this Talk) For given c , n ≥ 1, we combine linear functions with lookups in tables of size √ n filled with random values. c z ( i ) � h i ( x ) = f i ( x ) ⊕ j [ g j ( x ) ] , i = 1 , 2 j =1 Class of all these pairs ( h 1 , h 2 ) of hash functions : Z . (Extension of hash functions from (Dietzfelbinger/Woelfel ’03)) M. Aum¨ uller Graphs Generated by Simple Hash Functions 7/17
Example: Cuckoo Hashing with a Stash Main Task For given S and stash size s , calculate Pr(( h 1 , h 2 ) unsuitable for S with stash size s ) . Minimal bad subgraph: MOS s . (Example: s = 2.) M. Aum¨ uller Graphs Generated by Simple Hash Functions 8/17
Thus, we have ( h 1 , h 2 ) ∈Z (( h 1 , h 2 ) unsuitable for S with stash size s ) Pr = ( h 1 , h 2 ) ∈Z ( ∃ T ⊆ S : G ( T , h 1 , h 2 ) forms a MOS s ) Pr � ≤ ( h 1 , h 2 ) ∈Z ( G ( T , h 1 , h 2 ) forms a MOS s ) Pr T ⊆ S if ( h 1 , h 2 ) are fully random, we provide a direct counting argument that this is O (1 / n s +1 ) giving an alternative proof to the original analysis by Kirsch, Mitzenmacher and Wieder (who used machinery like Markov chain coupling) M. Aum¨ uller Graphs Generated by Simple Hash Functions 9/17
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